CHAPTER IV. 



EXAMPLES OF SOLENOIDAL FIELDS AND THEIR REPRESENTATION BY 



PLANE DRAWINGS. 



118. Two-Dimensional Representations of Three-Dimensional Vector-Fields. 

 Before passing to practical applications, it will be useful to consider a few simple 

 examples of solenoidal fields and to illustrate different methods of representing 

 them by plane drawings. 



In order to see the character of two-dimensional drawings representing any- 

 three-dimensional field, let us consider a surface cutting through the field in space. 

 At every point of the surface the vector has a certain direction and intensity. For 

 the representation it will be convenient to consider separately two projections of 

 the vector, that on the normal to the surface, and that on the plane tangential 

 to the surface. The normal component can be represented simply by curves for equal 

 numerical values. No representation of the direction is required. The field of this 

 component has lost the character of a vector-field and has completely obtained 

 that of a two-dimensional scalar field. 



The tangential component, on the other hand, will represent a true two-dimen- 

 sional vector-field. The methods of representing it geometrically will be special cases 

 of the methods for representing vectors in space (section 1 10) . Precisely as in space, 

 the direction can be represented by vector-lines. But instead of surfaces we shall get 

 curves of equal intensity. It should be observed that these curves of equal intensity 

 will not, as a rule, be the intersections of the given surface with the surfaces of equal 

 intensity in space. This will be the case only if the given surface happens to be a 

 vector-surface ; for then the normal vector will disappear and the vector of the two- 

 dimensional field will be identical with that of the three-dimensional field in space. 



A set of two-dimensional drawings representing a three-dimensional vector- 

 field in space can therefore be obtained in the following way: We choose a set of 

 surfaces cutting through the field. The vector defines at each of them a two- 

 dimensional vector- field and a two-dimensional scalar field. The first can be repre- 

 sented by two sets of curves, viz, the vector-lines and curves of equal intensity for 

 the tangential component; and the second by one set of curves, viz, curves for equal 

 values of the normal component. 



We shall then have to direct our attention to the two-dimensional vector-fields 

 contained in a surface and to the correlated scalar fields representing a vector- 

 component normal to the surface. 



1 19. General Remarks on the Two-Dimensional Vector-Field. For the same 

 reasons which we have for vector-lines in space, we get : 



Vector-lines of the two-dimensional field can intersect each other under finite 



angles only at points where the two-dimensional vector is zero, i. e., at 



points where the corresponding vector in space is either zero or normal 



to the surface containing the two-dimensional field. 



33 



